Extractions: Jump to: navigation search In mathematics category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics , and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942-1945, in connection with algebraic topology Category theory has several faces known not just to specialists, but to other mathematicians. " General abstract nonsense " refers, perhaps not entirely affectionately, to its high level of abstraction, compared to more classical branches of mathematics. Homological algebra is category theory in its aspect of organising and suggesting calculations in abstract algebra Diagram chasing is a visual method of arguing with abstract 'arrows', and has appeared in a Hollywood film, as Jill Clayburgh proved the snake lemma (at the start of It's My Turn Topos theory is a form of abstract sheaf theory , with geometric origins, and leads to ideas such as pointless topology Category theory Portal Background Historical notes Categories, objects and morphisms

Extractions: The Curry-Howard isomorphism edit this chapter This article attempts to give an overview of category theory, insofar as it applies to Haskell. To this end, Haskell code will be given alongside the mathematical definitions. Absolute rigour is not followed; in its place, we seek to give the reader an intuitive feel for what the concepts of category theory are and how they relate to Haskell. A simple category, with three objects A B and C , three identity morphisms i d A i d B and i d C , and two other morphisms and . The third element (the specification of how to compose the morphisms) is not shown. A category is, in essence, a simple collection. It has three components: A collection of objects A collection of morphisms , each of which ties two objects (a source object and a target object ) together. (These are sometimes called

Extractions: Log in Help Edit this page Discuss this page ... Related changes Category theory Categories Theoretical foundations map ... even maybeToList Just yields the same as maybeToList fmap even Just yields: both yield False In the followings, this example will be used to illustrate the notion of natural transformation. If the examples are exaggerated and/or the definitions are incomprehensible, try #External links edit Let us define the natural transformation. It associates to each object of a morphism of in the following way (usually, not sets are discussed here, but proper classes, so I do not use term âfunctionâ for this mapping): Thus, the following diagram commutes (in edit As already mentioned map even maybeToList Just yields the same as maybeToList fmap even Just yields: both yield False This example will be shown in the light of the above definition in the followings.

Extractions: the libarynth Trace: http://plato.stanford.edu/entries/category-theory/ Category theory is a general mathematical theory of structures and sytems of structures. It allows us to see, among other things, how structures of different kinds are related to one another as well as the universal components of a family of structures of a given kind. The theory is philosophically relevant in more than one way. For one thing, it is considered by many as being an alternative to set theory as a foundation for mathematics. Furthermore, it can be thought of as constituting a theory of concepts. Finally, it sheds a new light on many traditional philosophical questions, for instance on the nature of reference and truth. http://math.ucr.edu/home/baez/categories.html âBasic Category Theory for Computer Scientistsâ, Benjamin C. Pierce âA Categorical Manifestoâ by Goguen http://citeseer.nj.nec.com/goguen91categorical.html http://www.let.uu.nl/esslli/Courses/barr-wells.html http://scienceblogs.com/goodmath/goodmath/category_theory/

Category_theory « Rss2go Full Circle the Categorical Monoid Good Math, Bad Math. a category built using categories. The monoidal category is a fairly complicated object http://www.rss2go.net/topic/category_theory